I'm not familiar with MySQL but your statement looks strange (I'm not saying it's wrong). What's strange to me is you specify what column to use for join, but not what table this column belongs to. Then if you have 2 tables with the same column it can't work. Hopfully ANSI SQL is supported by MySQL then one of the following could work:
FROM products AS p
INNER JOIN categories AS c ON c.cid = p.cid
INNER JOIN merchants AS m ON m.mid = p.mid
WHERE c.categories = 'television'
FROM products AS p, categories AS c, merchants AS m
WHERE c.cid = p.cid
AND m.mid = p.mid
AND c.categories = 'television'